Chip-firing game and partial Tutte polynomial for
Eulerian digraphs††thanks: This paper was partially sponsored by Vietnam Institute for Advanced Study in Mathematics (VIASM) and the Vietnamese National Foundation for Science and Technology Development (NAFOSTED)
The Chip-firing game is a discrete dynamical system played on a graph, in which chips move along edges according to a simple local rule. Properties of the underlying graph are of course useful to the understanding of the game, but since a conjecture of Biggs that was proved by Merino López, we also know that the study of the Chip-firing game can give insights on the graph. In particular, a strong relation between the partial Tutte polynomial and the set of recurrent configurations of a Chip-firing game (with a distinguished sink vertex) has been established for undirected graphs. A direct consequence is that the generating function of the set of recurrent configurations is independent of the choice of the sink for the game, as it characterizes the underlying graph itself. In this paper we prove that this property also holds for Eulerian directed graphs (digraphs), a class on the way from undirected graphs to general digraphs. It turns out from this property that the generating function of the set of recurrent configurations of an Eulerian digraph is a natural and convincing candidate for generalizing the partial Tutte polynomial to this class. Our work also gives some promising directions of looking for a generalization of the Tutte polynomial to general digraphs.
Keywords. Chip-firing game, complexity, critical configuration, Eulerian digraph, feedback arc set, recurrent configuration, reliability polynomial, Sandpile model, Tutte polynomial.
There are insightful polynomials that are defined on undirected graphs, such as Tutte polynomial, chromatic polynomial, cover polynomial, reliability polynomial, etc, which evaluations count certain combinatorial objects. The Tutte polynomial is the most well-known, it has many interesting properties and applications [Tut53]. There is an evident interest in looking for analogues of the Tutte polynomial for directed graphs (digraphs), of for some other objects [Ges89, Gor93, CG95]. These attempts share properties of the Tutte polynomial. Nevertheless, they are not natural extensions of the Tutte polynomial in the sense that one does not know a conversion from the properties of these polynomials to those of the Tutte polynomial, in particular how to get back to the Tutte polynomial on undirected graphs from these polynomials. For this reason the authors of [CG95] asked for a natural generalization of Tutte polynomial for digraphs.
The evaluation of the Tutte polynomial at is important since it has a strong connection to the reliability polynomial that is studied in the network theory. In this paper we present a polynomial that can be considered as a natural generalization of for the class of Eulerian digraphs. An Eulerian digraph is a strongly connected digraph in which each vertex has equal in-degree and out-degree. An undirected graph can be regarded as an Eulerian digraph by replacing each edge by two reverse arcs and that have the same endpoints as . When considering undirected graphs seen as Eulerian digraphs in that way, we will see that we get back to the partial Tutte polynomial , which is a new and relevant feature.
This work is based on an idea conjectured by Biggs and proved by Merino Lópes, that the generating function of the set of recurrent configurations of the Chip-firing game of an undirected graph is equal to the partial Tutte polynomial [Big97, Lop97]. Based on a discrete dynamical system, this construction defines a polynomial that characterizes the graph supporting the dynamic. It is not straightforward to generalize those ideas to the class of Eulerian digraphs, but the results we will develop gives a promising direction for further extensions.
The Chip-firing game is a discrete dynamical system defined on a directed graph (digraph) , where some chips are stored on each vertex of . An assignment of chips on the vertices is called a configuration of , and a configuration can be transformed into a new configuration by the following rule: if a vertex of has as many chips as its out-degree and at least one out-going arc, then it is firable and the diffusion process called firing v consists in moving one chip of along each out-going arc to the corresponding vertex. The game playing with this rule is called Chip-firing game (CFG), and is called support graph of the game. A configuration is stable if it has no firable vertex. It is known that starting from any initial configuration the game either plays forever or converges to a unique stable configuration. If has a global sink, i.e., a vertex with out-degree and such that for any other vertex there is a path from to , then the game always converges for any choice of initial configuration [BLS91, BL92, HLMPPW08]. Throughout the paper we will concentrate on such CFG with a global sink. For a strongly connected digraph , we will choose a particular vertex and consider it as the sink by removing all out-going arcs of . The study of sink-independent properties (definitions that leads to the same object whatever vertex is chosen as the sink) will provide clues to define a natural analogue of the Tutte polynomial, for the class of Eulerian digraphs. The Chip-firing with a sink on digraphs has been introduced under the name Dollar game on undirected graphs.
The Dollar game is a variant of CFG on undirected graphs in which a particular vertex plays the role of a sink, and the sink can only be fired if all other vertices are not firable [Big99]. In this model the number of chips stored in the sink may be negative. This definition leads naturally to the notion of recurrent configurations (originally called critical configurations) that are stable, and unchanged under firing at the sink and stabilizing the resulting configuration. The definition of the Dollar game on Eulerian digraphs is the same as on undirected graphs, i.e. some vertex is chosen to be the sink that only can be fired only if all other vertices are not firable [HLMPPW08]. In the rest of the paper we will use the name Chip-firing game with a sink instead of Dollar game.
The set of recurrent configurations of a CFG with a sink on an undirected graph has many interesting properties, such as it is an Abelian group with the addition defined by vertex to vertex addition of chip content followed by stabilization, and its cardinality is equal to the number of spanning trees of the support graph, etc. Remarkably, Biggs defined the level of a recurrent configuration and made an intriguing conjecture about the relation between the generating function of recurrent configurations and the Tutte polynomial [Big97]. This conjecture was later proved by Merino López [Lop97]. A direct consequence is that the generating function of recurrent configurations of a CFG with a sink is independent of the chosen sink, and thus characterizes the support graph. This fact is definitely not trivial, and opened a new direction for studying graphs using the Chip-firing game as a tool [CB03, Mer05].
A lot of properties of recurrent configurations on undirected graphs can be extended to Eulerian digraphs without difficulty. However the situation is different when one tries to extend the sink-independence property of the generating function to a larger class of graphs, in particular to Eulerian digraphs, mainly because a natural definition of the Tutte polynomial for digraphs is unknown, nor is it for Eulerian digraphs. In this paper we develop a combinatorial approach, based on a level-preserving bijection between two sets of recurrent configurations with respect to two different sinks, to show that this sink-independence property also holds for Eulerian digraphs. This bijection provides new insights into the groups of recurrent configurations.
It turns out from the sink-independence property of the generating function, that this latter is a characteristic of the support Eulerian digraph, and we can denote it by regardless of the sink. We will see that evaluations of can be considered as extensions of to Eulerian digraphs, which make us believe that the polynomial is a natural generalization of . Furthermore, the most important feature is that and are equal on undirected graphs. It requires to be inventive to discover which objects the evaluations of counts, and we hope that further properties will be found. The class of Eulerian digraphs is in-between undirected and directed graph, and following the track we develop in this paper, we propose some conjectures that would be promising directions of looking for a natural generalization of to general digraphs.
The paper is divided into the following sections. Section 2 recalls known results on recurrent configurations on a digraph with global sink. Section 3 is devoted to the Eulerian digraph case, and Section 4 establishes the sink-independence of the generating function of recurrent configurations in that case. The Tutte polynomial generalization is presented in Section 5, and Section 6 hints at continuations of the present work.
2 Recurrent configurations on a digraph with global sink
All graphs in this paper are assumed to be multi-digraphs without loops. Graphs with loops will be considered in Section 5. We introduce in this section some notations and known results about recurrent configurations of CFG with a sink on general digraphs, followed by straightforward considerations on the number of chips stored on vertices of recurrent configurations.
For a digraph and an arc , we denote by and the tail and head of , respectively. For two vertices , let denote the number of arcs from to in . A configuration on is a map from to . A vertex is firable in if and only if . Firing a firable vertex is the process that decreases by and increases each with by . A sequence of vertices of is called a firing sequence of a configuration if starting from we can consecutively fire the vertices . Applying the firing sequence leads to configuration and we write , or without specifying the firing sequence.
In the rest of this section we assume that has a global sink . The definition of recurrent configurations is based on the convergence of the game, which is ensured if has a global sink. Since is not firable no matter how many chips it has, it makes sense to define a configuration to be a map from to . When a chip goes into the sink, it vanishes. The interest is to assimilate two configurations that have the same number of chips on every vertices except on the sink. Note that in this section we consider only one fixed sink, but in subsequent sections we will consider the CFG relatively to different choices of sink, and therefore we will need some more notations. Let us not be overburdened yet, a configuration on with sink is a map .
We recall a basic result of the Chip-firing game on digraphs with a global sink.
The following is simple but very important, and will often be used without explicit reference.
For two configurations and , we denote by the configuration given by for any . Then .
A stable configuration is recurrent if and only if for any configuration there is a configuration such that .
There are several equivalent definitions of recurrent configurations. The one above says that is recurrent if and only if it can be reached from any other configuration by adding some chips (according to ) and then stabilize.
Dhar proved that the set of recurrent configurations has an elegant algebraic structure [Dha90]. Fix a linear order on the vertices different from , where . Now a configuration of can be represented as a vector in . For each let be the vector in defined by if , otherwise if . Firing index then corresponds to adding the vector . We define a binary relation over by iff there exist such that , i.e. and are linked by a (possibly impossible to perform) sequence of firings. The following states the nice algebraic structure of the set of all recurrent configurations of with sink .
[HLMPPW08] The set of all recurrent configurations of is an Abelian group with the addition defined by . This group is isomorphic to . Moreover, each equivalence class of contains exactly one recurrent configuration, and the number of recurrent configurations is equal to the number of equivalence classes.
The group in Lemma 3 is called the Sandpile group of . The following simple properties can be derived easily from the definition of recurrent configuration.
The following holds
Let be a configuration such that for every . Then is recurrent.
Let and be two configurations such that for any . Then . Moreover, if is recurrent then is also recurrent.
For any configuration , adding grains according to leads to , and is a configuration with positive chip content on each vertex. Clearly, , therefore is recurrent.
Let be a firing sequence of such that . Since is the number of chips lost into the sink, we have . Since for any , is also a firing sequence of . Therefore there is a firing sequence of such that . For the same reason we have . The first claim follows.
Let be an arbitrary configuration. Since is recurrent, there is a configuration such that . Let be a configuration. We have , thus is recurrent.
3 Chip-firing game on an Eulerian digraph with a sink
Let be a digraph. The digraph is Eulerian if is connected and for every we have . With this condition the digraph is strongly connected. In this section we assume that is Eulerian, and present properties that recurrent configurations verify in that case.
As in the previous section, the definition of recurrent configuration is based on the convergence of the game. Therefore a global sink plays an important role in the definition. The digraph is strongly connected, therefore it has no global sink and the game may play forever from some initial configurations. To overcome this issue, we distinguish a particular vertex of that plays the role of the sink. Let be a vertex of , by removing all outgoing arcs of from we got the digraph that has a global sink . The Chip-firing game on with sink is the ordinary Chip-firing game that is defined on , and recurrent configurations are defined as presented above, on . Figures (a)a and (b)b present an example of and . It is a good way to think of the Chip-firing game on an Eulerian digraph with a sink as the ordinary Chip-firing game on with a fixed vertex that never fires in the game no matter how many chips it has. In this section we consider a fixed sink .
A configuration of the Chip-firing game on with sink is a map from to . To verify the recurrence of a configuration , we have to test the condition that for any configuration there is a configuration such that . This is a tiresome task. However, in the case of Eulerian digraphs we have the following useful criterion.
Figure (c)c presents a configuration . The configuration is presented in Figure (d)d, adding corresponds to firing the sink. To verify the recurrence of , one computes . Starting with we fire consecutively the vertices in this order and get exactly the configuration , therefore is recurrent. This procedure is called Burning algorithm. The following will be important later.
Let and be two stable configurations such that and are in the same equivalence class. If is recurrent then .
Let be the configuration that is defined as in Lemma 5 and be an arbitrary stable configuration. We claim that for any firing sequence of such that each vertex of occurs at most once in . For a contradiction we assume otherwise. This assumption implies that there is a first repetition, i.e., there is such that are pairwise-distinct and for some . We denote the configuration obtained from after the vertices have been fired. We will now show that is not firable in , a contradiction. Let be the number of chips receives from its in-neighbors when the vertices have been fired. Adding corresponds to firing the sink, thus . Since are pairwise-distinct and different from the sink, we have . The digraph is Eulerian, therefore and from the previous equality we have , but is stable so vertex is not firable in configuration , which is absurd.
Since each of the in-neighbors of is fired at most once in any firing sequence of such that , it follows that no more chips than that added to (that is ) can end up in the sink since is Eulerian, and consequently . Repeating the application of this inequality times we have , where is the configuration given by for any . This reasoning can be applied to and we have for any . Since for any vertex and any being an out-neighbor of there is a path in from to , with large enough we can add a sufficient number of chips so that there is an appropriate firing sequence of with and such that for any . When stabilizing , it follows from Lemma 1 (convergence), Lemma 4 (recurrence) and Lemma 3 (unicity of recurrent configuration in an equivalent class) that it leads to , that is, . This completes the proof. ∎
Does the claim of Lemma 6 hold for a general digraph with a global sink?
Note that if this statement is true, then it is tight. Figure 2 presents an example, on an undirected graph, of a recurrent configuration and a non-recurrent configuration belonging to the same equivalence class, such that they contain the same total number of chips. As a consequence, the recurrent configuration is not necessarily the unique configuration of maximum total number of chips over stable configurations of its equivalence class.
4 Sink-independence of generating function of recurrent configurations of an Eulerian digraph
Key observation. Let us give an important observation that motivates the study presented in this paper. We consider the Chip-firing game on the digraph drawn on Figure (a)a. In this game the vertex is chosen to be sink. All the recurrent configurations are presented in Figure 3. For each recurrent configuration we compute the sum of chips on the vertices different from the sink. We get the sorted sequence of numbers . If is chosen to be the sink of the game, all the recurrent configurations are given in Figure 4, and the sum of chips on vertices different from the sink gives the sorted sequence . The two sequences are the same up to adding a constant sequence. This property also holds with other choices of sink, therefore, up to a constant, this sequence is characteristic of the support graph itself. This interesting property is the main discovery exploited in this paper, and allows to generalize the construction presented in [Big97] and proved in [Lop97] of an analogue for the Tutte polynomial to the class of Eulerian digraphs. It is stated in the following theorem.
Let be an Eulerian digraph and a vertex of . For each recurrent configuration with respect to sink , let denote . The recurrent configurations with respect to sink are denoted by for some . Then the sequence is independent of the choice of up to a permutation of the entries.
The result of Merino López [Lop97] implies that Theorem 1 is true for undirected graphs. An undirected graph can be considered as an Eulerian digraph by replacing each undirected edge with two endpoints and by two reverse directed arcs and satisfying and . With this conversion it makes sense to call an Eulerian digraph undirected if for any two vertices of we have . The following known result is thus a particular case of Theorem 1, for the class of undirected graphs.
[Lop97] Let be the set of all recurrent configurations with respect to some sink . If is an undirected graph (defined as a digraph) then , where is the Tutte polynomial of and for any .
In the rest of this section we work with an Eulerian digraph . In order to prove Theorem 1, we consider the following natural approach. Let and be two distinct vertices of . We denote by and the sets of all recurrent configurations with respect to sink and , respectively. We are going to construct a bijection from to such that for every . Note that it follows from [HLMPPW08] that . In order to work on the CFG with respect to different sinks, we introduce some clear notations.
For a digraph , let denote a configuration that assigns a number of chips to every vertex, i.e., a map . In order to discard the number of chips stored in the sink , we introduce the notation , which is the map restricted to the domain . We denote the configuration obtained by stabilizing with respect to the sink . The operator can be applied to or and gives respectively a configuration or (note that since two operators and commute).
It follows from Lemma 1 that the configurations and are well-defined and unique. See Figure 5 for an illustration. A basic trick will be to stabilize according to a sink , choose a number of chips to assign to , and then stabilize according to another sink . Let us already state notations for this purpose.
Basically, we will add chips to the sink , so that it becomes firable. We therefore define as the configuration such that if and . Note that we will always apply this operator to a configuration on which the sink is specified.
Similarly, denotes the configuration where we put plus extra chips on , i.e. such that if and . Obviously, .
The sum of chips we are interested in may be applied to a configuration or , and is defined as .
The map is based on the following property, which describes the construction of a configuration belonging to from a configuration of . Note that the configurations of (resp. ) have type (resp. ) and are therefore denoted (resp. ). The procedure is straightforward: given a stable configuration of , we add chips to the sink and then stabilize according to the sink . The resulting configuration, restricted to , belongs to .
Let , then . Moreover, .
Since the concept of this lemma is at the heart of the construction of the map , we give an illustration of the claim before going into the details of the proof. We consider the Eulerian digraph given in Figure 1 with and . Figure (a)a shows a recurrent configuration with respect to sink . The configuration is given in Figure (b)b. Figure (c)c and Figure (d)d show and its restriction to . Using the Burning algorithm, one easily checks that the configuration in Figure (d)d is indeed recurrent with respect to sink .
Proof of Lemma 7.
We will once again use Lemma 5 (the Burning algorithm), which provides a firing sequence associated to a recurrent configuration of , that we will manipulate to built a firing sequence associated to a recurrent configuration of . In , only is firable, and after firing it, we will use the firing sequence leading back to , provided by Lemma 5.
Let (resp. ) be given by (resp. ) is equal to (resp. ). Let be such that . We have , and can therefore applyLemma 5 (since is recurrent with respect to sink ), providing a firing sequence of such that and each vertex of occurs exactly once in this sequence. Let be such that and be the configuration reached after vertices have been fired. Vertex is firable in , thus . Since the game is convergent, and this intermediate configuration is reachable from without firing , when we stabilize with respect to sink we end up with at least as many chips in as in , and the second part of the lemma follows.
We now prove that is a recurrent configuration of , by constructing a firing sequence in order to apply Lemma 5. Since , the sequence is a firing sequence of . We now consider the Chip-firing game with respect to sink (that is, on ), and the configuration . Let be such that . We have , and the rest of the firing sequence implies that , therefore for any . The recurrence of with respect to follows from the argument presented in the proof of Lemma 6. ∎
Lemma 7 naturally suggests a bijection from to that is defined by . However, this does not give the intended bijection since it does not necessarily preserves the sum of chips, as shown on Figure 6. The generalization of , denoted and corresponding to adding some extra chips to , is more flexible and can be used to improve the above map so that it preserves the sum of chips. That is what we are going to present now.
For all and any , we have . Moreover, .
Lemma 8 produces a map from to defined by
This map is injective from the following result:
Let and , then .
For convenience, let denote . It follows from Lemma 8 that restricted to belongs to and . As a consequence, some firings happen when stabilizing according to the sink . Let us prove that this process leads to . From Lemma 8 (with the application to ), we have . Since is Eulerian, the configurations and belong to the same equivalence class, so do and . Both are recurrent, hence from Lemma 3 they are equal. Finally, and obviously contain the same total number of chips, and are equal on the vertices different from , consequently they also contain the same number of chips on . ∎
The aim is now to find, for every recurrent configuration , the good so as to get a bijection from to that preserves the sum of chips. We first concentrate on the sum conservation: if one wants to have
then the number must be chosen so that , because the extra chips are not counted in both sum in this case. The following shows that such an always exists.
For every there exists such that .
Let the function be defined by . We are going to prove that there exists such that . Since for every , it follows from Lemma 4 (using the trick ) that , therefore . As a consequence for every , that is, the function decreases by at most one.
By Lemma 8 we have , therefore . Since for any , the proof is completed by showing that there is such that . In particular, we are going to prove that , where . Note that is the order of the Sandpile group of with respect to sink .
. We are going to use Lemma 6, which states that the recurrent configuration has maximum total number of chips over stable configurations of its equivalence class, in order to upper bound by , and the result follows since vertex is stable in the recurrent configuration (meaning that ).
Let be given by if and . We have (note that is a configuration since our digraph is Eulerian and has a global sink), thus the choice of implies that and are in the same equivalence class with respect to sink , and the first contains more chips than the latter. The configuration is recurrent, hence from Lemma 6 we have . It remains to exploit the total number of chips difference between the two configurations: . Replacing by on both sides, the inequality given by Lemma 6 thus becomes , and equivalently . ∎
We can now construct the intended bijection . For each , let denote the smallest number such that . The positive integer is called the swap number of from to . By Lemma 10 we know that swap numbers are well-defined and unique.